Special Values and Transcendence

主要研究者提出了算术几何和超越数论的几个项目，重点是来自特殊函数值的分析和算术信息之间的深层联系。 在一个项目中，研究者计划继续研究与安德森-德林费尔德动机相关的积极特征的量，包括周期，周期，伽马值，zeta值，多重zeta值和L值。 使用伽罗瓦理论的差分方程，他将追求结果的代数独立性，这些数量都单独和组合。在另一个项目中，研究者将考虑将函数域算术不变量与Goss-Hecke L-级数的特殊值相关联的新问题。 最后，研究者将探讨有限域超几何函数与Siegel模形式相关的L-级数的相关问题。数论是数学的基本分支之一，它是许多应用的基础，包括密码学和编码理论。 拟议的研究考虑的问题涉及的解析函数的值，有点显着传达基本信息领域的代数数或几何对象定义在他们身上。 这些问题起源于欧拉和高斯的工作，数学家们继续奋进解开它们的奥秘。 该项目的几个部分自然会给研究生和本科生的研究带来问题。

The principal investigator proposes several projects in arithmetic geometry and transcendental number theory that focus on the deep links between the analytic and arithmetic information coming from values of special functions. In one project, the investigator plans to continue the study of quantities related to Anderson-Drinfeld motives in positive characteristic, including periods, logarithms, Gamma values, zeta values, multiple zeta values, and L-values. Using the Galois theory of difference equations, he will pursue results about the algebraic independence of these quantities both individually and in combinations. In another project, the investigator will consider new problems which associate function field arithmetic invariants to special values of Goss-Hecke L-series. Finally, the investigator will pursue problems relating finite field hypergeometric functions to L-series associated to Siegel modular forms.Number theory is one of the fundamental branches of mathematics, and it serves as the basis for many applications, including cryptography and coding theory. The proposed research considers questions involving values of analytic functions that somewhat remarkably convey fundamental information about fields of algebraic numbers or geometric objects defined over them. Such questions have their genesis in work of Euler and Gauss, and mathematicians continue to endeavor to unravel their mysteries. Several parts of the project lead naturally to problems for graduate and undergraduate research.